Sub-wavelength waveguide loaded by a complementary electricmetamaterial for vacuum electron devices

Zhaoyun Duan,1,2,a) Jason S. Hummelt,1 Michael A. Shapiro,1,b) and Richard J. Temkin1

1Plasma Science and Fusion Center, Massachusetts Institute of Technology, 167 Albany St., Cambridge,Massachusetts 02139, USA2Institute of High Energy Electronics, School of Physical Electronics, University of Electronic Science andTechnology of China, No.4, Section 2, North Jianshe Road, Chengdu 610054, China

(Received 9 August 2014; accepted 25 September 2014; published online 9 October 2014)

We report the electromagnetic properties of a waveguide loaded by complementary electric split

ring resonators (CeSRRs) and the application of the waveguide in vacuum electronics. The S-

parameters of the CeSRRs in free space are calculated using the HFSS code and are used to retrieve

the effective permittivity and permeability in an effective medium theory. The dispersion relation

of a waveguide loaded with the CeSRRs is calculated by two approaches: by direct calculation

with HFSS and by calculation with the effective medium theory; the results are in good agreement.

An improved agreement is obtained using a fitting procedure for the permittivity tensor in the effec-

tive medium theory. The gain of a backward wave mode of the CeSRR-loaded waveguide interact-

ing with an electron beam is calculated by two methods: by using the HFSS model and traveling

wave tube theory; and by using a dispersion relation derived in the effective medium model.

Results of the two methods are in very good agreement. The proposed all-metal structure may be

useful in miniaturized vacuum electron devices. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4897392]

I. INTRODUCTION

Since a left-handed medium was first realized in 2000,1

researchers worldwide have been interested in its exotic electro-

magnetic properties, such as negative refractive index,2,3 back-

ward Cerenkov radiation,2,4 and the realization of different types

of metamaterials (MTMs) and their potential applications.

Rectilinear electron beams interacting with slow waves

in periodic structures are used in traveling wave tubes

(TWTs) or backward wave oscillators (BWOs). An interac-

tion circuit of a TWT or BWO can be designed as a helix,

coupled-cavity structure, folded waveguide, or ladder struc-

ture depending on the frequency range.5–7 MTMs and pho-

tonic crystals built of metallic parts can be considered as

modified coupled-cavity or ladder structures and are promis-

ing for application in TWTs or BWOs. The wave-electron

beam interaction in MTMs has been theoretically studied in

Refs. 8–15. Here, we focus on the MTM-loaded waveguide

and its application as an interaction circuit for a vacuum

electron device.

Waves in the MTM-loaded waveguide can be described

using different models. The MTM-loaded rectangular wave-

guide was theoretically studied as early as 2001.16

Subsequently, Marques et al. reported a split ring resonator

(SRR)-loaded waveguide and verified its transmission char-

acteristics in experiment when the waveguide operates below

the cutoff frequency of the TE mode.17 It has been shown

that the cutoff waveguide provides negative permittivity and

the SRRs offer negative permeability, which therefore

explains wave propagation in the SRR-loaded waveguide.

Esteban et al. showed that a rod-loaded waveguide supports

an electromagnetic wave propagating at frequencies, which

are below the cutoff frequency of the TM mode in the empty

waveguide.18 This is an example of wave propagation in a

waveguide that provides negative permeability because the

TM mode is below cutoff and the waveguide is loaded with

a negative permittivity medium. It was shown in Ref. 19 that

wave propagation in a MTM-loaded below-cutoff waveguide

can be explained by the anisotropy of the SRR medium. The

theory of a below-cutoff waveguide loaded by anisotropic

MTMs was also presented in Refs. 20 and 21.

In this paper, we study a closed metallic waveguide of

square cross-section loaded by metallic complementary

SRRs (CSRRs). This waveguide may have an application as

an interaction circuit for a vacuum electron device in which

a rectilinear electron beam excites a TM-dominated mode

with the longitudinal electric field in the direction of the

electron velocity.13 The CSRR is needed because it provides

electric response for the longitudinal electric field.13 In con-

trast to Ref. 13, in this paper, we consider a different type of

CSRR—planar complementary electric split ring resonators

(CeSRRs) introduced in Ref. 22 and we present a simple

model for a waveguide loaded by CeSRRs.

Using this example of a MTM-loaded waveguide, we

show that it is possible to determine the effective permittiv-

ity and permeability of the MTM, then fill the waveguide

with this effective medium and predict the waveguide

mode’s dispersion properties by using the constitutive pa-

rameters of the effective medium. The effective constitutive

parameters allow us to analyze the wave-beam interaction in

the MTM-loaded waveguide.

In Sec. II, we retrieve the effective constitutive parame-

ters for a CeSRR MTM in free space using the existing

a)zhyduan@uestc.edu.cnb)shapiro@psfc.mit.edu

1070-664X/2014/21(10)/103301/6/$30.00 VC 2014 AIP Publishing LLC21, 103301-1

PHYSICS OF PLASMAS 21, 103301 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.24.27.64 On: Fri, 31 Oct 2014 21:28:21

method of MTM constitutive parameter retrieval.23–28 In

Sec. III, we use the effective medium theory to study the dis-

persion characteristics of a cutoff metallic waveguide loaded

by the CeSRRs. Then we compare the effective medium

model with an HFSS simulation of an actual CeSRR loaded

waveguide. In Sec. IV, we discuss the application of the

MTM-loaded waveguide in vacuum electron devices.

II. COMPLEMENTARY ELECTRIC SPLIT RINGRESONATORS IN FREE SPACE

We first consider a MTM in free space, which consists of

planar CeSRRs. It produces a “negative” electric response and

not a “negative” magnetic response.22 In this paper, we use a

planar CeSRR, a schematic of which is shown in Fig. 1. The

dimensions shown in Fig. 1 refer to a specific design to oper-

ate near 3 GHz. The CeSRR has some attractive advantages

over the eSRR such as: greater symmetry that eliminates the

magnetoelectric response, which occurs in conventional

SRRs; an all-metal construction to avoid dielectrics in vac-

uum; an enhanced transmission due to the resonance; and easy

fabrication and assembly. If the dimension of the unit cell is

much less than the wavelength k, then the CeSRR MTM can

be considered as an effective medium. Accordingly, we can

retrieve the effective MTM constitutive parameters.

As a specific example, we choose the dimensions of the

unit cell as shown in Fig. 1. The periodic array of CeSRRs

with the period a is perforated on a plate. Identical plates are

set at distance a parallel to each other. We assume a plane

wave propagating in the y-direction and with polarization in

the z-direction, as shown in Fig. 1. The plane wave is nor-

mally incident on the CeSRR MTM surface. Here, we do not

consider the ohmic loss of the metal. Thus, the relative per-

mittivity and permeability are only real.

We set up the HFSS model (a cubic geometry of size

a� a� a) and use the driven-modal solver to obtain the S

parameters. The amplitude and phase of the S parameters are

shown in Figs. 2(a) and 2(b), respectively. The resonant

enhanced transmission at �3 GHz results from circulating

currents in the CeSRR and a pure electric response for the Ez

polarization. The counter-circulating currents eliminate any

magnetoelectric response.22

We use the S parameter-based retrieval method25 to

retrieve the effective permittivity ezz and permeability lxx.

The results are presented in Fig. 3(a). As noted in Ref. 22,

the CeSRR permittivity is the sum of an effective Drude-like

response (with an effective plasma frequency fp¼ 2.82 GHz

for the structure of Fig. 1), arising from the interconnected

metallic regions, and, at higher frequency, a Lorentz-like

response (with resonant frequency �3.35 GHz for the struc-

ture of Fig. 1). As for the permeability, in fact, there is no

magnetic response, as stated in Refs. 22 and 28.

By changing the polarization of the incident wave from

Ez to Ex (Fig. 1), we determine the S parameters and then

retrieve the permittivity exx and permeability lzz, as illus-

trated in Fig. 3(b).

III. COMPLEMENTARY ELECTRIC SPLIT RINGRESONATOR-LOADED CUTOFF WAVEGUIDE

A CeSRR-loaded cutoff waveguide is formed as fol-

lows. The CeSRR unit cell is periodically arranged along the

z-axis to form a plate (the period is a), which is positioned in

the middle of a square waveguide of cross-section a� a, as

shown in Fig. 4.

The square waveguide was simulated using the HFSS

code. Figure 5(a) shows the longitudinal field Ez distribution

in several axial slices of one period of the waveguide. The

CeSRR is in the middle of the cell at y¼ a/2. The transverse

distribution of the magnitude of Ez(y) is plotted in Fig. 5(b)

FIG. 1. The CeSRR unit cell used in this study. For operation near 3 GHz,

the structural parameters (in mm) are a¼ 14.5, b¼ 13.5, d¼ 1, j¼ 1.5,

h1¼ 4.25, h2¼ 4, g¼ 1, and thickness t¼ 1 (not shown here).

FIG. 2. The amplitude (a) and the phase (b) of the scattering parameters vs.frequency.

103301-2 Duan et al. Phys. Plasmas 21, 103301 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.24.27.64 On: Fri, 31 Oct 2014 21:28:21

for the frequency of 3.07 GHz. The field is evanescent near

the CeSRR plate and is zero at the wall. The walls (at y¼ 0

and y¼ a) do not significantly affect the field distribution

because the field amplitude and its derivative are small near

the walls. In fact, the HFSS simulation of the dispersion

characteristics indicates that the dispersion does not change

when the E-wall boundary condition is changed to the H-

wall boundary condition.

In Sec. II, the permittivity tensor of the medium consist-

ing of a periodic set of CeSRR plates was determined. The

MTM can be considered homogenized after the effective me-

dium parameters are determined. A planar waveguide can be

formed when the MTM is placed in between the metallic

walls perpendicular to the CeSRR plates, i.e., at x¼ 0 and

x¼ a. Therefore, one period of the MTM is placed in the x-

direction. In the y-direction, the CeSRR plates are set peri-

odically with the period a. The modes of this planar wave-

guide can be determined using the permittivity tensor and

the metal wall boundary conditions.

A. Dispersion equation of a planar MTM-loadedwaveguide

Now we formulate the wave equation for the modes of

the planar waveguide filled with a homogeneous effective

medium. We derive the dispersion equation for the TM

mode assuming that the field does not depend on y, because

the MTM is homogeneous.

The permittivity tensor is determined for the waves

propagating perpendicular to the CeSRR plates. Therefore, it

can be used to determine the waveguide modes close to cut-

off. The effective permittivity is anisotropic and the perme-

ability is isotropic, which leads to the following forms:

��e ¼ e0

exx 0 0

0 1 0

0 0 ezz

0@

1A; l ¼ l0; (1)

where e0 and l0 are the permittivity and permeability in vac-

uum, respectively.

Assuming an eixt time dependence (x is the angular fre-

quency) and a source free CeSRR-loaded cutoff waveguide

region, Maxwell’s equations can be written as

FIG. 3. Retrieved permittivities and permeabilities of the CeSRRs as func-

tions of frequency for z (a) and x (b) polarizations, respectively.

FIG. 4. The square metallic waveguide loaded with the CeSRR plate in the

middle. Here, a¼ 14.5 mm.

FIG. 5. The HFSS simulation of the square metallic waveguide loaded with

the CeSRR plate in the middle at y¼ 7.25 mm: (a) Ez field amplitude distri-

bution in one cell at the slices parallel to the CeSRR at y¼ 2.25, 4.5, 10, and

12.25 mm; (b) Ez field amplitude as a function of y; Ez is averaged over x.

103301-3 Duan et al. Phys. Plasmas 21, 103301 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.24.27.64 On: Fri, 31 Oct 2014 21:28:21

r� �E ¼ �ixl �H ; (2a)

r� �H ¼ ix�e• �E: (2b)

Meanwhile, we assume that the wave propagates along

the z-axis with an e�ibz dependence on z, where b is the

phase constant. In addition, we suppose that the mode has a

TM polarization. Inserting Eq. (1) into Eq. (2), making use

of the separation of variables procedure, we derive the scalar

wave equation for the Ez component as follows:

b2

x2exx=c2 � b2þ 1

!@2Ez

@x2þ x2

c2ezzEz ¼ 0; (3)

where c is the speed of light in free space. The boundary con-

ditions can be applied directly to Ez:

Ezðx; zÞ ¼ 0; at x ¼ 0; a: (4)

The resulting dispersion equation of the CeSRR-loaded cut-

off waveguide can be expressed as

p=að Þ2

ezzþ b2

exx¼ x2

c2; (5)

where ezz and exx are dependent on x.

B. Dispersion simulation

For the CeSRR-loaded sub-wavelength waveguide of

Fig. 4, we have simulated the exact dispersion curve using

the HFSS eigenmode solver and the result is the curve la-

beled “Simulation” in Fig. 6. We may also obtain the disper-

sion relation using the effective medium model, as described

in Sec. III A and Eq. (5). The resulting dispersion relation is

shown in Fig. 6 as the “Theory” curve. Both approaches are

in good qualitative agreement; both predict that the existing

wave is a backward wave, which results from the negative

exx and positive ezz, not a forward wave.

From this comparison of the HFSS simulation and the

effective medium model, we conclude that it is possible to

retrieve the MTM constitutive parameters and predict the

dispersion of the MTM-loaded waveguide. The retrieved exx

and ezz can be used in the dispersion equation (Eq. (5)) to cal-

culate the dispersion of the mode close to cutoff when the

phase advance is much smaller than p and the periodicity in

the z-direction does not affect the dispersion.

C. Dispersion relation with fitting ezz and exx

Good qualitative agreement is obtained between the dis-

persion relations of the effective medium model and the

HFSS simulation. We can obtain better quantitative agree-

ment by using fitted Drude models of the relative permittiv-

ities exx and ezz in Eq. (5) using the form

exx ¼ Mxxð1� x2pxx=x

2Þ; ezz ¼ Mzzð1� x2pzz=x

2Þ: (6)

Here, Mxx and Mzz are constants. The plasma frequencies are

xpxx/2p¼ 4.2 GHz and xpzz/2p¼ 2.835 GHz. The constants

selected for a best fitting to the HFSS dispersion are Mxx¼ 1.5

and Mzz¼ 32. The result of calculating the dispersion relation,

Eq. (5), using the relative permittivities of Eq. (6) is shown in

Fig. 6 as the curve labeled “Fitting.” This curve is in very

good agreement with the HFSS simulation results.

Below a frequency of 3.0 GHz, the guide wavelength

(kg¼ 2p/b) is not very much larger than the unit cell size,

which means the effective medium theory begins to deviate

from the exact result. This fact is evidenced by the disagree-

ment between the HFSS simulated dispersion curve and the

dispersion curve using the fitting ezz and exx as the phase

advance exceeds p/2 (Fig. 6).

The backward mode is unidirectional; it can support a

fast wave (left of the light line) and a slow wave (right of the

light line), which is useful for vacuum electron devices.

IV. APPLICATION IN A VACUUM ELECTRON DEVICE

The sub-wavelength MTM waveguide is suitable for a

vacuum electron device. It can benefit vacuum electron devi-

ces where a small structure size is needed.13,29

We have shown that the simple model described by Eq.

(5) is in agreement with the HFSS simulation of the MTM

waveguide. The MTM waveguide mode can be excited by

the rectilinear electron beam. The wave-beam interaction

occurs in the vicinity of the intersection of the wave disper-

sion curve and the beam dispersion line x¼ bV, V is the ve-

locity of the electron beam. In Fig. 6, the beam dispersion

line is shown for the beam voltage of 100 kV.

A. Gain equation for effective medium theory

Using the effective medium permittivity tensor compo-

nents ezz and exx, we determine the gain of the Cerenkov

instability of the electron beam in the MTM waveguide.13

The new longitudinal permittivity ezzn including the electron

beam is given by30

FIG. 6. Dispersion curves of Frequency vs. Phase Advance (ba) calculated

by Simulation (HFSS), theory (Eq. (5)) and fitting (Eqs. (5) and (6)). The

dispersion curves are for the backward wave mode in the CeSRR-loaded

square waveguide. The light line is x¼bc and the electron beam line is

x¼bV, where the frequency is x/2p, and V is the velocity of the electron

beam.

103301-4 Duan et al. Phys. Plasmas 21, 103301 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.24.27.64 On: Fri, 31 Oct 2014 21:28:21

ezzn ¼ ezz �x2

b

x� bVð Þ2; (7)

where the beam plasma frequency is determined by

x2b ¼ e2nb=e0m; nb is the electron density, e and m are the

electron charge and mass, respectively. The relativistic

effects are not taken into account in this formulation. Eq. (5)

thus can be rewritten as

p=að Þ2

ezznþ b2

exx¼ x2

c2: (8)

From Eqs. (6) and (8), we derive the wave-beam interaction

dispersion equation

b2 � b20

� �b� x

V

� �2

¼ �x2b

V2

p2

a2

exx

e2zz

; (9)

where b0 is the solution of Eq. (8) with xb¼ 0. Eq. (9) can

be recognized as having the form of the dispersion relation

of a traveling wave tube.31 By neglecting interaction with

the counter-propagating wave b¼�b0 and assuming no

detuning from synchronism between the wave and the beam

(b0¼x/V), we determine the maximum of the imaginary

part of the propagation constant:

Imb ¼ffiffiffi3p

2

xV� p2x2

bV2

2x4a2

exx

e2zz

!1=3

: (10)

Equation (10) can be rewritten for the gain G as

G ¼ 2Imb ¼ffiffiffi3p x

V� I

I0

2p3 c3V

x4a4

exx

e2zz

!1=3

; (11)

where I0¼ 4pe0mc3/e� 17 kA.

B. Gain calculation using HFSS code

The gain in Eq. (11) was determined by using the effec-

tive medium representation of the CeSRRs to derive the dis-

persion relation. We may compare that result with a second,

independent calculation of the gain using the HFSS code to

calculate the wave-beam coupling impedance. The coupling

impedance is then used in traveling wave tube theory to

obtain the gain. The mean coupling impedance is represented

as follows:31

�Kc ¼

1S

Ð Ðs

jEzj2ds

2b2P; (12)

where P is the electromagnetic power propagating in the

waveguide and Ez is the longitudinal electric field, which is

averaged over the electron beam cross-section (S¼ a� a).

The Pierce parameter C is determined by

C ¼ I �Kc

4U

� �1=3

; (13)

using the beam current I and the beam voltage U. The gain

of the wave-beam instability is31

G ¼ffiffiffi3p x

VC: (14)

Fig. 7 shows a comparison of the gain calculation using

the two approaches: the effective medium approach (Eq.

(11)) versus a rigorous calculation using the exact fields cal-

culated with the HFSS code. The gain is proportional to I1/3,

therefore, the ratio G/I1/3 is plotted as a function of the fre-

quency. The voltage U of the synchronous beam is varied

with the frequency. The gain results of the simple effective

medium theory model using the fitting parameters (Sec. III

C) are in very good agreement with those of the HFSS simu-

lation (Fig. 6). Meanwhile, Fig. 7 also shows the frequency

tuning of the gain of the backward wave as the beam line

(x¼bV, see Fig. 6) varies with a change of voltage.

The Cerenkov instability of the electron beam in the

MTM has specific features. The gain is higher as compared

to a conventional Cerenkov electron device because the

group velocity of the wave is lower. A backward-wave oscil-

lator can be built based on this instability because the group

velocity of the wave is negative.

V. CONCLUSION

We have characterized the CeSRR MTMs and proposed

a CeSRR-loaded cutoff waveguide. The effective permittiv-

ity and permeability have been retrieved using the S

parameter-based retrieval method for the CeSRR MTM in

free space. Meanwhile, we have studied the wave propaga-

tion in a cutoff waveguide loaded by the CeSRRs using both

theory and simulation. A backward wave mode has been

found in the MTM-loaded waveguide. The theoretical dis-

persion relation derived for the CeSRR MTM, which is con-

sidered as an effective medium agrees qualitatively with the

numerical simulation. Very good agreement between the

HFSS simulations and the effective medium model has been

achieved by using the fitting permittivity tensor.

The interaction of the MTM waveguide wave with an

electron beam has been studied. The coupling with the beam

was calculated using HFSS and compared to the gain pre-

dicted by the effective medium model, with very good

FIG. 7. Comparison of wave-beam instability Gain as a function of frequency

between the HFSS simulation using Eq. (14) and the effective medium theory

based on the fitting permittivity using Eq. (11). The current I is in Amperes.

103301-5 Duan et al. Phys. Plasmas 21, 103301 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.24.27.64 On: Fri, 31 Oct 2014 21:28:21

agreement. The effective medium approach is found to be

useful for calculation of the modes of the waveguides loaded

by the MTM. The gain of the wave-beam interaction in the

MTM-loaded waveguide can be analyzed using the effective

medium parameters. This novel MTM structure may allow

for the miniaturization of vacuum electron devices.

ACKNOWLEDGMENTS

Work supported by AFOSR MURI Grant FA9550-12-1-

0489 administered by the University of New Mexico; DOE,

High Energy Physics (Grant No. DE-SC0010075); and the

Natural Science Foundation of China (Grant Nos. 61471091

and 61125103).

1D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz,

Phys. Rev. Lett. 84, 4184 (2000).2V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).3G. Shvets, Phys. Rev. B 67, 035109 (2003).4Z. Y. Duan, C. Guo, and M. Chen, Opt. Express 19, 13825 (2011).5A. S. Gilmour, Jr., Klystrons, Traveling Wave Tubes, Magnetrons, Cross-Field Amplifiers, and Gyrotrons (Artech House, Boston-London, 2011).

6Y. M. Shin, G. S. Park, G. P. Scheitrum, and B. Arfin, IEEE Trans. Plasma

Sci. 31, 1317 (2003).7C. L. Kory, M. E. Read, R. L. Ives, J. H. Booske, and P. Borchard, IEEE

Trans. Electron Devices 56, 713 (2009).8Y. P. Bliokh, S. Savel’ev, and F. Nori, Phys. Rev. Lett. 100, 244803

(2008).9D. Shiffler, R. Seviour, E. Luchinskaya, E. Stranford, W. Tang, and D.

French, IEEE Trans. Plasma Sci. 41, 1679 (2013).10M. I. Bakunov, R. V. Mikhaylovskiy, S. B. Bodrov, and B. S.

Luk’yanchuk, Opt. Express 18, 1684 (2010).11D. Shiffler, J. Luginsland, D. M. French, and J. Watrous, IEEE Trans.

Plasma Sci. 38, 1462 (2010).

12Z. Y. Duan, C. Guo, J. Zhou, J. C. Lu, and M. Chen, Phys. Plasmas 19,

013112 (2012).13M. A. Shapiro, S. Trendafilov, Y. Urzhumov, A. Al�u, R. J. Temkin, and G.

Shvets, Phys. Rev. B 86, 085132 (2012).14Z. Y. Duan, C. Guo, X. Guo, and M. Chen, Phys. Plasmas 20, 093301

(2013).15D. M. French, D. Shiffler, and K. Cartwright, Phys. Plasmas 20, 083116

(2013).16C. Caloz, C. C. Chang, and T. Itoh, J. Appl. Phys. 90, 5483 (2001).17R. Marqu�es, J. Martel, F. Mesa, and F. Medina, Phys. Rev. Lett. 89,

183901 (2002).18J. Esteban, C. Camacho-Pe~nalosa, J. E. Page, T. M. Mart�ın-Guerrero,

and E. M�arquez-Segura, IEEE Trans. Microw. Theory Tech. 53, 1506

(2005).19I. G. Kondratyev and A. I. Smirnov, Radiophys. Quantum Electron. 48,

136 (2005).20P. A. Belov and C. R. Simovski, Phys. Rev. E 72, 036618 (2005).21S. Hrabar, J. Bartolic, and Z. Sipus, IEEE Trans. Antenn. Propag. 53, 110

(2005).22H. T. Chen, J. F. O’Hara, A. J. Taylor, R. D. Averitt, C. Highstrete, M.

Lee, and W. J. Padilla, Opt. Express 15, 1084 (2007).23D. R. Smith, S. Schultz, P. Marko�s, and C. M. Soukoulis, Phys. Rev. B 65,

195104 (2002).24X. Chen, T. M. Grzegorczyk, B. I. Wu, J. Pacheco, and J. A. Kong, Phys.

Rev. E 70, 016608 (2004).25D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, Phys. Rev. E

71, 036617 (2005).26J. T. Costa, M. G. Silveirinha, and S. I. Maslovski, Phys. Rev. B 80,

235124 (2009).27Z. Szab�o, G. H. Park, R. Hedge, and E. P. Li, IEEE Trans. Microw.

Theory Tech. 58, 2646 (2010).28D. R. Smith, Phys. Rev. E 81, 036605 (2010).29Z. Y. Duan, B. I. Wu, S. Xi, H. S. Chen, and M. Chen, Prog. Electromagn.

Res. PIER 90, 75 (2009).30S. T. Ivanov, O. V. Dolgenko, and A. A. Rukhadze, J. Phys. A: Math. Gen.

8, 585 (1975).31S. E. Tsimring, Electron Beams and Microwave Vacuum Electronics (John

Wiley and Sons, New Jersey, 2007).

103301-6 Duan et al. Phys. Plasmas 21, 103301 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.24.27.64 On: Fri, 31 Oct 2014 21:28:21